April 13, 2005
Lecturer Dmitri Zaitsev
Hilary Term 2005
Course 2E1 2004-05 (SF Engineers & MSISS & MEMS)
S h e e t 19
Due: in the tutorial sessions first Wednesday/Thursday in the next term
Exercise 1
Find the characteristic polynomials of the following matrices:
4 0
(i)
;
0 −3
0 1
(ii)
;
2 0


0 1 1
(iii)  0 2 2 ;
0 0 3


0 1 1
(iv)  0 2 2 .
0 2 3
Exercise 2
Find the eigenvalues and the corresponding eigenvectors of the following matrices:
2 0
(i)
;
1 −3


1 1 2
(ii)  0 2 4 ;
0 2 4
Exercise 3
Find a matrix
0
(i) A =
4

2
(ii) A =  0
0
P that diagonalizes the given matrix A and determine D = P −1 AP :
2
;
0

0 −2
3 0 .
0 3